Abstract: Brizolis asked the question: does every prime have a pair such that is a fixed point for the discrete logarithm with base ? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on and . In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

8.Mark
Kac, Statistical independence in probability, analysis and number
theory., The Carus Mathematical Monographs, No. 12, Published by the
Mathematical Association of America. Distributed by John Wiley and Sons,
Inc., New York, 1959. MR
0110114

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